Invariants
Level: | $56$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.288 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&39\\55&18\end{bmatrix}$, $\begin{bmatrix}32&19\\33&24\end{bmatrix}$, $\begin{bmatrix}44&53\\23&8\end{bmatrix}$, $\begin{bmatrix}50&47\\13&54\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
Full 56-torsion field degree: | $129024$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 14 x^{2} + 15 y^{2} - 26 y z + 15 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.12.0.t.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
28.12.0.n.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
56.12.0.bk.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
56.48.1.fb.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.fh.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.gf.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.gp.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.hl.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.hv.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.jb.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.jh.1 | $56$ | $2$ | $2$ | $1$ |
56.192.11.gp.1 | $56$ | $8$ | $8$ | $11$ |
56.504.34.jx.1 | $56$ | $21$ | $21$ | $34$ |
56.672.45.lb.1 | $56$ | $28$ | $28$ | $45$ |
168.48.1.bhj.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bhp.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bin.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bix.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.cbp.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.cbz.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.ccx.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.cdd.1 | $168$ | $2$ | $2$ | $1$ |
168.72.4.yz.1 | $168$ | $3$ | $3$ | $4$ |
168.96.3.tx.1 | $168$ | $4$ | $4$ | $3$ |
280.48.1.bej.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.bep.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.bfn.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.bfx.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.bvv.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.bwf.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.bxd.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.bxj.1 | $280$ | $2$ | $2$ | $1$ |
280.120.8.if.1 | $280$ | $5$ | $5$ | $8$ |
280.144.7.ud.1 | $280$ | $6$ | $6$ | $7$ |
280.240.15.bhv.1 | $280$ | $10$ | $10$ | $15$ |